How to Calculate Square Roots in Your Head

Learn how square roots are used in the real world, how you can easily estimate the value of a square root in your head, and how you can use an ancient algorithm to calculate the value of a square root by hand to as high a precision as needed.

If you’ve read last week’s article, figuring out what the square root of 16, 81, and every other number that’s a perfect square should be easy since you know that you just have to find the number that was squared to create the perfect square. Figuring out what the square root of a negative number like –1 or –42 should be easy as well since we saw that negative numbers don’t actually have square roots (at least none that are any numbers we know about yet). And figuring out what the square root of a non-perfect square like π or 55 should also be a piece of cake since you can always use a calculator.

But what if you don’t have a calculator handy when you really need to find that square root? What can you do? Keep on reading because today we’ll learn how to easily calculate square roots in your head.

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Recap: What Are Square Roots?

Before we get into the details of how to estimate square roots in your head and how to accurately calculate their values without a calculator, let’s quickly recap what square roots are. The idea is actually very simple: Whenever you multiply a number by itself—and keep in mind this goes for any number whether it’s an integer, rational, or irrational—that’s called squaring the number.

As an example with integers 15 x 15 is the square of 15, which is equal to 225. When we find the square root of the number 225, all we’re doing is figuring out what that original number was that we multiplied by itself to get 225. When dealing with numbers that aren’t perfect squares, the mechanics of finding square roots is different, but the idea is still the same. For example, what’s the square root of 343? Using a calculator, I find that the answer is approximately 18.52. If we multiply 18.52 x 18.52, we get 342.99 which is really close to 343. So, as you can see, no matter what type of number we’re dealing with, the meaning of the square root is the same.